Laying a Brick Walkway

Date: 04/22/2002 at 09:25:38
From: Dawn
Subject: Math Word Problem
I have to build a walkway, 2ft. by 20ft., out of bricks that are 1ft.
by 2ft. The bricks can lie vertically and horizontally but in no
other direction.
How many different ways can I build this walkway?
I know there has to be a pattern, but I do not see it. Any help would
be great.
Dawn

Date: 04/22/2002 at 12:30:51
From: Doctor Peterson
Subject: Re: Math Word Problem
Hi, Dawn.
When I don't see how to solve a problem like this, I start by playing
with the ideas involved in the problem. That usually means
simplifying the problem by working with smaller, easier numbers -
actually, that's more or less what children's play is, isn't it? By
trying things out in a smaller case, I get a feel for how it works.
And in this case, that leads directly to a solution.
How many ways can you make a 2-by-1 walkway (silly as it sounds!)?
One, of course:
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How about a 2-by-2 walkway? Two ways:
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How about 2-by-3? You can build on what you have already done: if the
first brick you place is vertical (across the walk), you have 2 more
feet to fill in, using one of the solutions for the 2-by-2. What
happens if the first brick is horizontal (lengthwise)? See if you can
use this to identify a pattern.
You will recognize a relationship to the Fibonacci sequence, which
you can read about in the Dr. Math FAQ:
Golden Ratio, Fibonacci Sequence
http://mathforum.org/dr.math/faq/faq.golden.ratio.html
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/